Quantum divergences, channel discrimination and strong data-processing

在经典信息理论和量子信息理论中，信息论分歧(也称为相对熵)是大量操作相关量的母量。它们可以用来表示熵(数据压缩的相关量)、互信息(经典数据传输和信道模拟中的相关量)和量化区分任务中状态的可区分性。理解这些分歧的数学特性通常会立即转化为在相关的操作问题上取得进展。虽然经典信息论中的分歧已经被研究了很长时间，并且基本上被完全分类和理解，但在量子环境中，由于不同的算符顺序，存在着无限多的可能性，并且不断引入新的有用的变体，这为研究开辟了新的道路。通过利用量子信息论的最新进展提供的数学工具和框架，我们希望进一步研究量子发散的性质，并特别关注这两个领域的问题：1.量子通道的熵量：发散作为区分任务中的相关可分辨度量，尽管状态区分的理论已经被很好地理解，但关于信道的区分(即态之间的物理转换)的许多问题仍然是悬而未决的。这些又常常与底层通道分叉的数学属性联系在一起，这些分叉涉及对输入状态的优化，因此更难处理。这里的一个重要问题是找出何时(以及如何具体地)基于先前的通道输出来选择哪些通道被顺序使用以及哪些输入到连续通道使用的区分策略可以提供比通道并行使用的更简单的策略更好的优势。后者与航道分流的“链条”性质有关。我们想要详细研究这一点，目的是充分了解并行和顺序战略是如何相关的，以及有时如何存在两者之间的优势。2.强数据处理：量子发散的定义(也可以说是最重要的)属性之一是数据处理不等性：只有当一对状态受到处理操作(即量子通道的作用)时，它们的可区分性才会降低。强大的数据处理涉及找到分辨率严格降低的条件，理想情况下还会限制这种降低。最近已经得到了几个关于发散差的下界，这些下界取决于通道的作用是否可以近似逆转。事实证明，这在随后的许多应用程序中都非常有用。这些界限是否紧，以及紧到什么程度，仍然是一个悬而未决的问题，有一些迹象表明，往往不是，我们希望对此进行调查。除了限制差异，我们还希望查看对输入状态进行优化的通道应用前后的差异的商的限制。这些所谓的收缩系数与快速混合(即开放量子系统指数级快速收敛到其平衡态)这一非常实际的问题密切相关，尤其与量子动力学半群(即与无记忆环境相互作用的系统的量子演化)有关。虽然这些收缩系数在经典上是很好理解的，但从量子力学的角度来看，仍然有许多悬而未决的问题，而且已经找到了收缩系数小于1的简单充分条件，这将是许多应用向前迈出的一大步。

Information-theoretic divergences (also called relative entropies) serve as parent quantities for a plethora of operationally relevant quantities both in classical and quantum information theory. They can be used to express entropy (relevant quantity for data-compression), mutual information (relevant quantity in classical data transmission and channel simulation) and quantify distinguishability of states in discrimination tasks. Understanding mathematical properties of these divergences often immediately translates to making progress on related operational questions. While divergences in classical information theory have been long studied and are essentially fully classified and understood, in the quantum setting, due to different operator orderings, there are infinitely many more possibilities, and new useful variants keep getting introduced, which open up new pathways in research. By building on mathematical tools and frameworks provided by recent advances in quantum information theory, we want to further investigate properties of quantum divergences, and specifically focus on these two areas of questions:1. Entropic quantities for quantum channels: Divergences serve as relevant distinguishability measures in discrimination tasks, and while the theory of state discrimination is very well understood, many questions regarding discrimination of channels (i.e. physical transformations between states) are still open. These are again often tied to mathematical properties of the underlying channel divergences which involve an optimization over input states and are hence trickier to deal with. One important question here is to find out when (and how specifically) discrimination strategies in which channels are used sequentially, and in which inputs to successive channel uses, are chosen based on previous channel outputs can give an advantage over simpler strategies, in which channels are used in parallel. The latter is related to a "chain-rule" property of the channel divergence. We want to study this in detail with the aim of fully understanding how parallel and sequential strategies are related and how there can sometimes be an advantage of one over the other. 2. Strong data processing: One of the defining (and arguably most important) properties of a quantum divergence is the data processing inequality: distinguishability of a pair of states can only decrease if they are subjected to processing operation (i.e. the action of a quantum channel). Strong data processing involves finding conditions under which distinguishability strictly decreases, and ideally also bound this decrease. Recently several lower bounds have been obtained on the difference of the divergences, which depend on whether the action of the channel can be approximately reversed. This has turned out to be very useful in many subsequent applications. It is still an open question if and to what degree these bounds are tight, with some indication that they are often not, which we would like to investigate. Instead of bounding the difference, we also want to look at bounding the quotient of the divergences before and after application of a channel, optimized over input states. These so-called contraction coefficients are closely related to very practical questions of rapid mixing (i.e. exponentially fast convergence of an open quantum system towards its equilibrium state), especially relevant in the context of quantum dynamical semigroups (i.e. quantum evolution of a system interacting with a memoryless environment). While classically these contraction coefficients are well understood, quantum-mechanically there are still many open questions, and already finding simple sufficient conditions for a contraction coefficient to be less than one would be a major step forward for many applications.